Consider a Mukai vector of the form \(v = (0, 2\theta , \chi ) \in H^{ev}_{alg}(J, \mathbb {Z})\)  with \(\chi \ne 0\) . Sheaves on \(J\)  with such a Mukai vector are pure of dimension one. Let \(\mathbf {M}_v\)  be the moduli space of semi-stable sheaves on \(J\)  with Mukai vector \(v\) . By [1]}, \(\mathbf {M}_v\)  is a normal irreducible projective variety of dimension 10 and the smooth locus, which is precisely the locus parameterizing strictly stable sheaves, admits a symplectic form. There is a natural morphism
\(\mathbf {a}_v \colon \mathbf {M}_v \rightarrow J^\vee \times J; \quad [{F}] \mapsto \big (\det ({F}) \otimes \mathcal {O}_J(2\theta )^\vee , \textstyle \sum c_2({F})\big ),\) 
